For the critical long-range percolation metric, geodesics are almost surely unique and every geodesic has Hausdorff dimension equal to the distance exponent theta.
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Uniqueness and dimension for the geodesic of the critical long-range percolation metric
For the critical long-range percolation metric, geodesics are almost surely unique and every geodesic has Hausdorff dimension equal to the distance exponent theta.